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Remarks on the r and Δ convolutions

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Abstract

In this paper we study the properties of the r–deformation introduced in [B1]. We observe that the associated convolution coming from the conditionally free convolution is associative only for r = 1 and r = 0. We give the realization of some r–Gaussian random variables and obtain Haagerup–Pisier–Buchholz type inequalities. We also study another convolution defined with the use of the r–deformation through a moment–cumulant formula [KY1] and show that it is associative and in general not positive.

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Authors and Affiliations

  1. Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Marek Bożejko, Anna Dorota Krystek & Łukasz Jan Wojakowski

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  1. Marek Bożejko
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  2. Anna Dorota Krystek
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  3. Łukasz Jan Wojakowski
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Correspondence to Marek Bożejko.

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Partially sponsored with KBN grant no 2P03A00723 and RTN HPRN-CT-2002-00279.

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Bożejko, M., Krystek, A. & Wojakowski, Ł. Remarks on the r and Δ convolutions. Math. Z. 253, 177–196 (2006). https://doi.org/10.1007/s00209-005-0898-2

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  • DOI: https://doi.org/10.1007/s00209-005-0898-2

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